Let `f(x) =sin^4x+cos^4x.` Then f is increasing function in the interval

Options

A. `(pi/4,pi/2)`
B. `((5pi)/8,(3pi)/4)`
C. `(0,pi/4)`
D. `(pi/2,(5pi)/8)`

Explanation

To determine the intervals where the function \( f(x) = \sin^4 x + \cos^4 x \) is increasing, we first need to find the derivative \( f'(x) \) and analyze its sign.

### Step 1: Find the derivative \( f'(x) \)

We start by differentiating \( f(x) \):

\[ f(x) = \sin^4 x + \cos^4 x \]

Using the chain rule, we differentiate each term:

\[ f'(x) = 4\sin^3 x \cdot \cos x - 4\cos^3 x \cdot \sin x \]

This can be factored as:

\[ f'(x) = 4\sin x \cos x (\sin^2 x - \cos^2 x) \]

### Step 2: Set the derivative to zero

To find the critical points, we set \( f'(x) = 0 \):

\[ 4\sin x \cos x (\sin^2 x - \cos^2 x) = 0 \]

This gives us two cases:

1. \( \sin x = 0 \)
2. \( \sin^2 x - \cos^2 x = 0 \)

### Step 3: Solve for critical points

**Case 1:** \( \sin x = 0 \)

This occurs at:

\[ x = n\pi \quad (n \in \mathbb{Z}) \]

**Case 2:** \( \sin^2 x - \cos^2 x = 0 \)

This simplifies to:

\[ \sin^2 x = \cos^2 x \implies \tan^2 x = 1 \implies \tan x = \pm 1 \]

This occurs at:

\[ x = \frac{\pi}{4} + n\frac{\pi}{2} \quad (n \in \mathbb{Z}) \]

### Step 4: Analyze the sign of \( f'(x) \)

We need to analyze the intervals determined by the critical points \( x = n\pi \) and \( x = \frac{\pi}{4} + n\frac{\pi}{2} \).

1. **Interval \( (0, \frac{\pi}{4}) \)**:
- Choose \( x = \frac{\pi}{8} \):
- \( \sin x > 0 \) and \( \cos x > 0 \), so \( f'(x) > 0 \).

2. **Interval \( (\frac{\pi}{4}, \frac{\pi}{2}) \)**:
- Choose \( x = \frac{\pi}{3} \):
- \( \sin x > 0 \) and \( \cos x > 0 \), but \( \sin^2 x < \cos^2 x \), so \( f'(x) < 0 \).

3. **Interval \( (\frac{\pi}{2}, \frac{3\pi}{4}) \)**:
- Choose \( x = \frac{5\pi}{8} \):
- \( \sin x > 0 \) and \( \cos x < 0 \), so \( f'(x) > 0 \).

4. **Interval \( (\frac{3\pi}{4}, \pi) \)**:
- Choose \( x = \frac{7\pi}{8} \):
- \( \sin x > 0 \) and \( \cos x < 0 \), but \( \sin^2 x < \cos^2 x \), so \( f'(x) < 0 \).

### Conclusion

From the analysis, we find that \( f(x) \) is increasing in the intervals \( (0, \frac{\pi}{4}) \) and \( (\frac{\pi}{2}, \frac{3\pi}{4}) \).